×

zbMATH — the first resource for mathematics

A new numerical approach to solve an elliptic equation. (English) Zbl 1084.65104
Summary: A new numerical approach to solve an elliptic partial differential equation that originates from the governing equations of steady state fluid flow and heat transfer is presented. The elliptic partial differential equation is transformed by introducing an exponential function to eliminate the convection terms in the equation. A fourth-order central differencing scheme and a second-order central differencing scheme are used to numerically solve the transformed elliptic partial differential equation. Analytical solutions of this equation are also given.
Comparisons are made between the analytical solutions, the numerical results using the present schemes, and those using the four classical differencing schemes, namely, the first-order upwind scheme, hybrid scheme, power-law scheme, and exponential scheme. The comparisons illustrate that the proposed algorithm performs better than the four classical differencing schemes.

MSC:
65N06 Finite difference methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Patankar, S.V., Numerical heat transfer and fluid flow, (1980), Hemisphere Washington, DC · Zbl 0595.76001
[2] Price, H.S.; Varga, R.S.; Warren, J.E., Application of oscillation matrices to diffusion-correction equations, Journal of mathematics and physics, 45, 301-311, (1996) · Zbl 0143.38301
[3] Raithby, G.D., Skew upstream differencing schemes for problems involving fluid flow, Computer methods in applied mechanics and engineering, 9, 153-164, (1976) · Zbl 0347.76066
[4] Leonard, B.P., A stable and accurate convective modeling procedure based on quadratic upstream interpolation, Computer methods in applied mechanics and engineering, 19, 59-98, (1979) · Zbl 0423.76070
[5] Leonard, B.P., Simple high-accuracy resolution program for convective modeling of discontinuities, International journal for numerical methods in fluids, 8, 1291-1318, (1988) · Zbl 0667.76125
[6] Xu, H.; Zhang, C., Numerical solution of a transformed parabolic equation, Applied mathematics and computation, 139, 535-554, (2003) · Zbl 1027.65117
[7] Chen, C.J., Finite analytical methods, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.